Instructor: Professor Yuri Bakhtin
Lectures: Mondays, Wednesdays 3:30-4:45PM CI/WWH 102
Office hours: Tentatively, Thursday 3-5pm at my office CI/WWH 729
Prerequisites: The course will routinely use a variety of tools from undergraduate analysis such as limits, series, Taylor expansions, partial derivatives, multiple integrals. We will also use basics of linear algebra and complex numbers. Modern probability theory is based on measure theory. Acquaintance with undegraduate probability and measure theory is a plus but it will not be assumed. The most important informal requirement is to be ready to study rigorous mathematics and proofs.
Books: There is no single book I would follow very closely. For reference, I will mostly use the following two books:
[K] Probability by Davar Khoshnevisan available electronically through NYU at https://www.ams.org/books/gsm/080/gsm080.pdf
[JP] Probability Essentials by Jean Jacod and Philip Protter available electronically through NYU at https://link-springer-com.proxy.library.nyu.edu/book/10.1007/978-3-642-55682-1
At various previous times this course was offered, the following books were also recommended by various instructors including myself (alphabetical order):
[D] Probability: Theory and Examples, by Richard Durrett https://www.cambridge.org/core/books/probability/DD9A1907F810BB14CCFF022CDFC5677A
[GS] Probability and Random Processes, by Geoffrey Grimmett and David Stirzaker, Oxford University Press
[V] Probability Theory, by S.R.S Varadhan, vol 7, Courant Lecture Notes in Mathematics, AMS https://www.ams.org/books/cln/007/cln007.pdf
[W] Knowing the Odds: An Introduction to Probability (Graduate Studies in Mathematics, 139) by John Walsh, although I am not going to follow it too closely. It is available electronically through NYU: https://www-ams-org.proxy.library.nyu.edu/books/gsm/139/gsm139.pdf
Course outline: This course introduces basic concepts and methods of probability theory and some applications. It is based on measure theory and is meant to be (mostly) rigorous. No prior knowledge of probability or measure theory is assumed. The plan is to cover most of the topics in the books [K],[JP], with varying level of detail: probability spaces, random variables, distributions, independence, expectations, conditional expectations, notions of convergence, Law of Large Numbers, Central Limit Theorem, characteristic functions, Markov chains, random walks, martingales. If time permits, we will study basics of the Wiener process aka Brownian motion.
Homework assignments will be collected every week or two. They will count for 20% of the final score. The lowest homework score will be dropped.
Two midterm exams will be given in class on Wednesday March 5 and Wednesday April 9. They will count for 20% of the final score each.
Final exam will count for 40% of the final score.
Study recommendations: I expect students to study the lecture notes and/or the relevant material in the recommended books between classes. The complexity of the material grows and each class (or book chapter) depends on the previous classes/chapters. Start working on homework assignments early, preferably as soon as they are available. If you have questions, please attend my office hours or ask your questions by email (do not expect immediate replies though).
Communication: Outside of classroom, I will mostly communicate with you through Announcements in Brightspace. You can reach me by email at bakhtin@cims.nyu.edu. Most of you are familiar with the basics of efficient email communication but here is a couple of useful links: https://cims.nyu.edu/~bakhtin/email-etiquette.html